ON THE PARAMETERISED COMPLEXITY OF CUTTING A FEW VERTICES FROM A - - PowerPoint PPT Presentation

ON THE PARAMETERISED COMPLEXITY OF CUTTING A FEW VERTICES FROM A GRAPH

Fedor

• V. Fomin1

Petr A. Golovach1 Janne H. Korhonen*,2

1University of Bergen, Norway 2University of Helsinki, Finland

INTRODUCTION MIN CUTS AND SIZE CONSTRAINTS

MINIMUM CUT

Given a graph G = (V,E), find a cut (X,Y) such that |∂(X,Y)| is minimised

MINIMUM CUT

Given a graph G = (V,E), find a cut (X,Y) such that |∂(X,Y)| is minimised

X Y

MINIMUM CUT

Given a graph G = (V,E), find a cut (X,Y) such that |∂(X,Y)| is minimised

Polynomial-time solvable! X Y

MINIMUM BISECTION

Given a graph G = (V,E), find a cut (X,Y) such that |∂(X,Y)| is minimised and |X| = |Y|

MINIMUM BISECTION

Given a graph G = (V,E), find a cut (X,Y) such that |∂(X,Y)| is minimised and |X| = |Y|

?

MINIMUM BISECTION

Given a graph G = (V,E), find a cut (X,Y) such that |∂(X,Y)| is minimised and |X| = |Y|

? NP-hard!

MINIMUM BISECTION ?

• More generally, we may require that...
• ...the smaller side of the cut has at most size k
• ...the smaller side of the cut has at least size k

PROBLEM DEFINITIONS PARAMETERISED SMALL CUTS

CUTTING A FEW VERTICES FROM A GRAPH

INPUT: Graph G = (V,E), positive integers k and t PROBLEM: Is there a cut (X,Y) such that

• 1≤ |X| ≤ k
• size of the cut is at most t

CUTTING A FEW VERTICES FROM A GRAPH

INPUT: Graph G = (V,E), positive integers k and t PROBLEM: Is there a cut (X,Y) such that

• 1≤ |X| ≤ k
• size of the cut is at most t
• Parameterised complexity perspective
• Assume k and t are small
• Variants
• Edge cuts vs. vertex cuts
• Terminals
• Size exactly k
• Parameterisation by k, by t, or by k + t

• Edge cut

Find a vertex partition (X,Y) s.t.

• 1≤ |X| ≤ k
• |∂(X,Y)| ≤ t
• Vertex cut

Find a vertex partition (X,S,Y) s.t.

• S separates X and Y
• 1≤ |X| ≤ k
• |S| ≤ t

CUTTING A FEW VERTICES FROM A GRAPH

X X S

∂(X,Y)

INPUT: Graph G = (V,E), positive integers k and t

BASIC VARIANTS

• Edge cut with terminal

Find a vertex partition (X,Y) s.t.

• |X| ≤ k
• |∂(X,Y)| ≤ t
• x ∈ X
• Vertex cut with terminal

Find a vertex partition (X,S,Y) s.t.

• S separates X and Y
• |X| ≤ k
• |S| ≤ t
• x ∈ X

CUTTING A FEW VERTICES FROM A GRAPH

X X S

∂(X,Y)

INPUT: Graph G = (V,E), positive integers k and t, vertex x

TERMINAL VARIANTS

x x

• Edge cut with exact size

Find a vertex partition (X,Y) s.t.

• |X| = k
• |∂(X,Y)| ≤ t
• Vertex cut with exact size

Find a vertex partition (X,S,Y) s.t.

• S separates X and Y
• |X| = k
• |S| ≤ t

CUTTING A FEW VERTICES FROM A GRAPH

X X S

∂(X,Y)

INPUT: Graph G = (V,E), positive integers k and t

EXACT VARIANTS

CUTTING A FEW VERTICES FROM A GRAPH

• edge cut
• edge cut, terminal
• edge cut, |X| = k
• vertex cut
• vertex cut, terminal
• vertex cut, |X| = k

CUTTING A FEW VERTICES FROM A GRAPH

• edge cut
• edge cut, terminal
• edge cut, |X| = k
• vertex cut
• vertex cut, terminal
• vertex cut, |X| = k
• For each of the variants above, is there an

algorithm with running time

• f(k)nO(1),
• f(t)nO(1), or
• f(k+t)nO(1)?

CUTTING A FEW VERTICES FROM A GRAPH

• edge cut
• edge cut, terminal
• edge cut, |X| = k
• vertex cut
• vertex cut, terminal
• vertex cut, |X| = k
• For each of the variants above, is there an

algorithm with running time

• f(k)nO(1),
• f(t)nO(1), or
• f(k+t)nO(1)?

i.e. FPT or W[1]-hard?

EARLIER WORK AND OUR RESULTS

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal edge cut, |X| = k vertex cut vertex cut, terminal vertex cut, |X| = k

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal edge cut, |X| = k vertex cut vertex cut, terminal vertex cut, |X| = k

polynomial

• Extreme sets [Watanabe and Nakamura 1987]
• Weighted version also in P [Armon and Zwick 2006]

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k vertex cut vertex cut, terminal vertex cut, |X| = k

polynomial

• [Lokshtanov and Marx 2011]

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

vertex cut vertex cut, terminal vertex cut, |X| = k

polynomial

• [Lokshtanov and Marx 2011]
• [Cai 2008]

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

vertex cut FPT vertex cut, terminal vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

• [Lokshtanov and Marx 2011]
• [Cai 2008]
• [Marx 2006]

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

vertex cut

W[1]-hard

FPT vertex cut, terminal W[1]-hard W[1]-hard FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

• Our results
• W[1]-hard results standard
• FPT result uses standard two-colour colour-

coding [Cai, Chan and Chan 2006]

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

• Our results
• main result uses important separators [Marx 2006]

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

• Colour-coding + knapsack [Cao 2013]

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

(open) FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

(open) FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

PROOF TECHNIQUES

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

(open) FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

(open) FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal W[1]-hard W[1]-hard FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

• W[1]-hardness results are standard

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

(open) FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

• Two-colour colour-coding [Cai, Chan and Chan 2006]

COLOUR-CODING

COLOUR-CODING

• Consider the basic vertex cut version

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red
• With probability ≥ 2-k-t we colour X blue and S red

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red
• With probability ≥ 2-k-t we colour X blue and S red
• Check all connected blue components

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red
• With probability ≥ 2-k-t we colour X blue and S red
• Check all connected blue components
• Repeat 2k+t nO(1) times

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red
• With probability ≥ 2-k-t we colour X blue and S red
• Check all connected blue components
• Repeat 2k+t nO(1) times
• Derandomisable using standard techniques

COLOUR-CODING

• Consider the basic vertex cut version
• Assume that there is a cut (X,S,Y) with |X| ≤ k and |S| ≤ t
• Colour each vertex randomly either blue or red
• With probability ≥ 2-k-t we colour X blue and S red
• Check all connected blue components
• Repeat 2k+t nO(1) times
• Derandomisable using standard techniques

Theorem. Vertex cut, vertex/edge cut with terminal and edge cut with |X| = k variants of cutting a few vertices problem can be solved in time 2k+t+o(k+t)nO(1).

compl mplexity parameter rameter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

(open) FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

• Main result of our paper
• uses important separators [Marx 2006]

IMPORTANT SEPARATORS, INTUITIVELY X Y

G = (V,E)

• Graph G = (V,E), subsets X and Y of V

IMPORTANT SEPARATORS, INTUITIVELY X Y

G = (V,E)

• Graph G = (V,E), subsets X and Y of V
• Basic idea: important (X,Y)-separator of size t is a

separator of size t that is as close to Y as possible

IMPORTANT SEPARATORS, FORMALLY X Y

G = (V,E)

• For a vertex set S, define R(X,S) as the set of vertices

reachable from X in G[V\S]

S R(X,S)

IMPORTANT SEPARATORS, FORMALLY X Y

G = (V,E)

• For a vertex set S, define R(X,S) as the set of vertices

reachable from X in G[V\S]

• Minimal (X,Y)-separator S is important if there is no (X,Y)-

separator T such that

• |T| ≤ |S|
• R(X,S) is a proper subset of R(X,T)

S R(X,S)

IMPORTANT SEPARATORS, FORMALLY X Y

G = (V,E)

• For a vertex set S, define R(X,S) as the set of vertices

reachable from X in G[V\S]

• Minimal (X,Y)-separator S is important if there is no (X,Y)-

separator T such that

• |T| ≤ |S|
• R(X,S) is a proper subset of R(X,T)

S R(X,S) T

IMPORTANT SEPARATORS, THEOREMS

Theorem [Marx 2006, Chen, Liu and Lu 2009]. There are at most 4t important (X,Y)-separators of size at most t, and they can be listed in time 4tnO(1).

IMPORTANT SEPARATORS, THEOREMS

Theorem [Marx 2006, Chen, Liu and Lu 2009]. There are at most 4t important (X,Y)-separators of size at most t, and they can be listed in time 4tnO(1). Theorem [Marx 2006]. There is exactly one important (X,Y)-separator of minimum size, and it can be found in polynomial time.

MAIN RESULT

Theorem. Vertex cut variant of cutting a few vertices problem can be solved in time 4tnO(1).

PROOF OF MAIN THEOREM IDEA 1.

G = (V,E)

PROOF OF MAIN THEOREM IDEA 1.

G = (V,E)

• Assume that solution (X,S,Y) exists

PROOF OF MAIN THEOREM IDEA 1.

G = (V,E)

• Assume that solution (X,S,Y) exists
• Guess vertices x ∈ X and y ∈

Y

x y

PROOF OF MAIN THEOREM IDEA 1.

G = (V,E)

• Assume that solution (X,S,Y) exists
• Guess vertices x ∈ X and y ∈

Y

• Test if some important (y,x)-separator is solution

x y

• Sadly, the idea does not work...

PROOF OF MAIN THEOREM IDEA 1.

• Sadly, the idea does not work...

PROOF OF MAIN THEOREM IDEA 1. x y

• Sadly, the idea does not work...

PROOF OF MAIN THEOREM IDEA 1. x y

• Sadly, the idea does not work...

PROOF OF MAIN THEOREM IDEA 1. x y

PROOF OF MAIN THEOREM IDEA 2.

G = (V,E)

PROOF OF MAIN THEOREM IDEA 2.

G = (V,E)

• Assume that solution (X,S,Y) exists

PROOF OF MAIN THEOREM IDEA 2.

G = (V,E)

x

• Assume that solution (X,S,Y) exists
• Guess a vertex x ∈ X

PROOF OF MAIN THEOREM IDEA 2.

G = (V,E)

x

• Assume that solution (X,S,Y) exists
• Guess a vertex x ∈ X
• STEP 1: use minimum-size important separators to find

vertices “far” away from x

PROOF OF MAIN THEOREM IDEA 2.

G = (V,E)

• Assume that solution (X,S,Y) exists
• Guess a vertex x ∈ X
• STEP 1: use minimum-size important separators to find

vertices “far” away from x

x Z

PROOF OF MAIN THEOREM IDEA 2.

G = (V,E)

• Assume that solution (X,S,Y) exists
• Guess a vertex x ∈ X
• STEP 1: use minimum-size important separators to find

vertices “far” away from x

• STEP 2: test if some important (Z,x)-separator is a solution

x Z

PROOF OF MAIN THEOREM IDEA 2. STEP 1 STEP 2

Z x x

PROOF OF MAIN THEOREM IDEA 2. STEP 1 STEP 2

• Problem: some vertices in Z may be inside X

Z x x

PROOF OF MAIN THEOREM IDEA 2. STEP 1 STEP 2

• Problem: some vertices in Z may be inside X
• Two possibilities (fortunately):
• Step 1 accidentally finds a solution (X’,S’,Y’) such that

x ∉ X’

• Otherwise there is only polynomial amount of

removals to try Z x x

PROOF OF MAIN THEOREM IDEA 2. STEP 1 STEP 2

• Problem: some vertices in Z may be inside X
• Two possibilities (fortunately):
• Step 1 accidentally finds a solution (X’,S’,Y’) such that

x ∉ X’

• Otherwise there is only polynomial amount of

removals to try

• Step 2 also requires k > t; otherwise, use colour-coding

Z x x

compl mplexity parameter meter variant k t k+t edge cut edge cut, terminal FPT FPT FPT edge cut, |X| = k

W[1]-hard

(open) FPT vertex cut

W[1]-hard

FPT FPT vertex cut, terminal

W[1]-hard W[1]-hard

FPT vertex cut, |X| = k

W[1]-hard W[1]-hard W[1]-hard

polynomial

Thank you!

Questions?